The document discusses complex numbers, including: - Complex numbers are based on the imaginary unit i, where i^2 = -1. - Complex numbers can be expressed as a + bi and graphed in the complex plane. - Operations like addition, subtraction, multiplication, and division can be performed on complex numbers by combining real and imaginary parts. - Quadratic equations can have complex number solutions. Finding these solutions involves factoring or using the quadratic formula.

1. 4-8 COMPLEX NUMBERS
Chapter 4 Quadratic Functions and Equations
©Tentinger

2. ESSENTIAL UNDERSTANDING AND
OBJECTIVES
 Essential Understanding: the complex numbers are
based on a number whose square is -1
 Objectives:
 Students will be able to:
 Identify complex numbers
 Graph and perform operations using complex numbers
 Find complex number solutions to quadratic equations

3. IOWA CORE CURRICULUM
 Number and Quantity
 N.CN.1. Know there is a complex number I such
that i2 = -1, and every complex number has the
form a + bi with a and b real.
 N.CN.2. Use the relation i2 = -1 and the
commutative, associative, and distributive
properties to add, subtract, and multiply complex
numbers.
 N.CN.7. Solve quadratic equations with real
coefficients that have complex solutions
 N.CN.8 (+) Expand polynomial identities to the
complex numbers

4. IMAGINARY NUMBERS
 What is an imaginary  Rewrite using the
number? imaginary unit i
 Represented as i  √-18
 i = √-1  √-12
 i2 = -1  √-7
 √-25
 √-24

5. IMAGINARY NUMBERS
 Imaginary number: any number in the form a + bi,
where a and b are real numbers and b does not
equal zero.
 Imaginary numbers and real numbers make up the
set of complex numbers

6. ABSOLUTE VALUE
 Absolute value of a complex number: the
distance from the origin in the complex plane

7. GRAPHING IN THE COMPLEX NUMBER PLANE
 The y-axis is the imaginary numbers, the x-axis is
the real numbers
 What are the graph and absolute values of each
number?
 5 –i
 -3 – 2i
 1 + 4i

8. ADDING AND SUBTRACTING
 You can define operations on the set of complex
numbers so that when you restrict the operations to
the subset of real numbers, you get the familiar
operations on the real numbers.
 Adding and Subtracting Complex Numbers
 Combine the real numbers and the imaginary
numbers
 (7-2i)+(-3+i)
 (8 + 6i)-(8-6i)
 (1+5i)-(3-2i)
 (-3+9i)+(3+9i)

9. MULTIPLYING
 Multiplying Complex Numbers
 Multiply as you would binomials
 (3i)(-5+2i)
 (7i)(3i)
 (2-3i)(4+5i)
 (-4+5i)(-4-5i)

10. DIVIDING
 Complex Conjugates: when (a+bi)(a-bi) = a real
number
 Use Complex Conjugates to simplify quotients

11. FACTORING
 Factoring Using Complex Conjugates
 2x2 + 32
 5x2 + 20
 x2 +81

12. IMAGINARY SOLUTIONS
 Every quadratic equation has complex number
solutions (that sometimes are real numbers)
 Finding Imaginary Solutions
 2x2 – 3x + 5 = 0
 3x2 – x + 2 = 0
 x2 – 4x + 5 = 0

13. HOMEWORK
 Pg. 253 – 254
 #9 – 69 (3s)